On the \(b\)-quark running mass in QCD and the SM. (English) Zbl 1356.81211

Bednyakov, A. V.; Kniehl, B. A.; Pikelner, A. F.; Veretin, O. L.

Nucl. Phys., B 916, 463-483 (2017).

Summary: We consider electroweak corrections to the relation between the running \(\overline{\mathrm{MS}}\) mass \(m_b\) of the \(b\) quark in the five-flavor \(\mathrm{QCD} \times \mathrm{QED}\) effective theory and its counterpart in the Standard Model (\(\mathrm{SM}\)). As a bridge between the two parameters, we use the pole mass \(M_b\) of the \(b\) quark, which can be calculated in both models. The running mass is not a fundamental parameter of the \(\mathrm{SM}\) Lagrangian, but the product of the running Yukawa coupling \(y_b\) and the Higgs vacuum expectation value. Since there exist different prescriptions to define the latter, the relations considered in the paper involve a certain amount of freedom. All the definitions can be related to each other in perturbation theory. Nevertheless, we argue in favour of a certain gauge-independent prescription and provide a relation which can be directly used to deduce the value of the Yukawa coupling of the \(b\) quark at the electroweak scale from its effective QCD running mass. This approach allows one to resum large logarithms \(\ln(m_b / M_t)\) systematically. Numerical analysis shows that, indeed, the corrections to the proposed relation are much smaller than those between \(y_b\) and \(M_b\).

MSC:

81V05	Strong interaction, including quantum chromodynamics
81V22	Unified quantum theories
81V15	Weak interaction in quantum theory
81T15	Perturbative methods of renormalization applied to problems in quantum field theory

Software:

mr; RunDec

References:

[1]	Kobayashi, M.; Maskawa, T., CP-violation in the renormalizable theory of weak interaction, Prog. Theor. Phys., 49, 652-657, (1973)
[2]	Herb, S. W., Observation of a dimuon resonance at 9.5 gev in 400-gev proton-nucleus collisions, Phys. Rev. Lett., 39, 252-255, (1977)
[3]	Bevan, A. J., The physics of the B factories, Eur. Phys. J. C, 74, 3026, (2014)
[4]	Fleischer, R., Theoretical prospects for B physics, PoS FPCP2015, (2015)
[5]	Pich, A., Effective field theory, Course, 949-1049, (1998)
[6]	Grozin, A. G., Heavy quark effective theory, Springer Tracts Mod. Phys., 201, 1-213, (2004)
[7]	Bigi, I. I.; Shifman, M. A.; Uraltsev, N.; Vainshtein, A., Pole mass of the heavy quark: perturbation theory and beyond, Phys. Rev. D, 50, 2234-2246, (1994)
[8]	Tarrach, R., The pole mass in perturbative QCD, Nucl. Phys. B, 183, 384-396, (1981)
[9]	Hoang, A. H.; Ligeti, Z.; Manohar, A. V., B decay and the ϒ mass, Phys. Rev. Lett., 82, 277-280, (1999)
[10]	Beneke, M., A quark mass definition adequate for threshold problems, Phys. Lett. B, 434, 115-125, (1998)
[11]	Pineda, A., Determination of the bottom quark mass from the \(\operatorname{\Upsilon}(1 S)\) system, J. High Energy Phys., 0106, (2001)
[12]	El-Khadra, A. X.; Luke, M., The mass of the b quark, Annu. Rev. Nucl. Part. Sci., 52, 201-251, (2002)
[13]	Peskin, M. E., Comparison of LHC and ILC capabilities for Higgs boson coupling measurements
[14]	Klute, M.; Lafaye, R.; Plehn, T.; Rauch, M.; Zerwas, D., Measuring Higgs couplings at a linear collider, Europhys. Lett., 101, 51001, (2013)
[15]	Hempfling, R.; Kniehl, B. A., On the relation between the fermion pole mass and \(\overline{\operatorname{MS}}\) Yukawa coupling in the standard model, Phys. Rev. D, 51, 1386-1394, (1995)
[16]	Kniehl, B. A.; Piclum, J. H.; Steinhauser, M., Relation between bottom-quark \(\overline{\operatorname{MS}}\) Yukawa coupling and pole mass, Nucl. Phys. B, 695, 199-216, (2004)
[17]	Kniehl, B. A.; Veretin, O. L.; Kniehl, B. A.; Veretin, O. L., Two-loop electroweak threshold corrections to the bottom and top Yukawa couplings, Nucl. Phys. B, Nucl. Phys. B, 894, 56-57, (2015), (Corrigendum) · Zbl 1328.81257
[18]	Marquard, P.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M., Quark mass relations to four-loop order, Phys. Rev. Lett., 114, (2015)
[19]	Bekavac, S.; Seidel, D.; Steinhauser, M.; Grozin, A., Light quark mass effects in the on-shell renormalization constants, J. High Energy Phys., 0710, (2007)
[20]	Bekavac, S.; Grozin, A. G.; Seidel, D.; Smirnov, V. A., Three-loop on-shell Feynman integrals with two masses, Nucl. Phys. B, 819, 183-200, (2009) · Zbl 1194.81252
[21]	Olive, K. A., Review of particle physics, Chin. Phys. C, 38, (2014)
[22]	Coleman, S.; Weinberg, E., Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D, 7, 1888-1910, (1973)
[23]	Jackiw, R., Functional evaluation of the effective potential, Phys. Rev. D, 9, 1686-1701, (1974)
[24]	Nielsen, N. K., On the gauge dependence of spontaneous symmetry breaking in gauge theories, Nucl. Phys. B, 101, 173-188, (1975)
[25]	Sirlin, A.; Zucchini, R., Dependence of the Higgs coupling \(\overline{h}_{\overline{\operatorname{MS}}}(M)\) on \(m_H\) and the possible onset of new physics, Nucl. Phys. B, 266, 389-409, (1986)
[26]	Actis, S.; Ferroglia, A.; Passera, M.; Passarino, G., Two-loop renormalization in the standard model. part I: prolegomena, Nucl. Phys. B, 777, 1-34, (2007) · Zbl 1200.81110
[27]	Actis, S.; Passarino, G., Two-loop renormalization in the standard model. part II: renormalization procedures and computational techniques, Nucl. Phys. B, 777, 35-99, (2007) · Zbl 1200.81111
[28]	Actis, S.; Passarino, G., Two-loop renormalization in the standard model. part III: renormalization equations and their solutions, Nucl. Phys. B, 777, 100-156, (2007) · Zbl 1200.81112
[29]	Fleischer, J.; Jegerlehner, F., Radiative corrections to Higgs-boson decays in the Weinberg-Salam model, Phys. Rev. D, 23, 2001-2026, (1981)
[30]	Denner, A.; Jenniches, L.; Lang, J.-N.; Sturm, C., Gauge-independent \(\overline{\operatorname{MS}}\) renormalization in the 2HDM, J. High Energy Phys., 1609, (2016)
[31]	Degrassi, G.; Di Vita, S.; Elias-Miró, J.; Espinosa, J. R.; Giudice, G. F.; Isidori, G.; Strumia, A., Higgs mass and vacuum stability in the standard model at NNLO, J. High Energy Phys., 1208, (2012)
[32]	Buttazzo, D.; Degrassi, G.; Giardino, P. P.; Giudice, G. F.; Sala, F.; Salvio, A.; Strumia, A., Investigating the near-criticality of the Higgs boson, J. High Energy Phys., 1312, (2013)
[33]	Martin, S. P.; Robertson, D. G., Higgs boson mass in the standard model at two-loop order and beyond, Phys. Rev. D, 90, (2014)
[34]	Martin, S. P., Pole mass of the W boson at two-loop order in the pure \(\overline{\operatorname{MS}}\) scheme, Phys. Rev. D, 91, (2015)
[35]	Martin, S. P., Z-boson pole mass at two-loop order in the pure \(\overline{\operatorname{MS}}\) scheme, Phys. Rev. D, 92, (2015)
[36]	Martin, S. P., Top-quark pole mass in the tadpole-free \(\overline{\operatorname{MS}}\) scheme, Phys. Rev. D, 93, (2016)
[37]	Bednyakov, A. V.; Pikelner, A. F.; Velizhanin, V. N., Three-loop Higgs self-coupling beta-function in the standard model with complex Yukawa matrices, Nucl. Phys. B, 879, 256-267, (2014) · Zbl 1284.81320
[38]	Jegerlehner, F.; Kalmykov, M. Yu.; Veretin, O., \(\overline{\operatorname{MS}}\) vs. pole masses of gauge bosons: electroweak bosonic two-loop corrections, Nucl. Phys. B, 641, 285-326, (2002)
[39]	Jegerlehner, F.; Kalmykov, M. Yu.; Veretin, O., \(\overline{\operatorname{MS}}\) vs. pole masses of gauge bosons II: two-loop electroweak fermion corrections, Nucl. Phys. B, 658, 49-112, (2003) · Zbl 1097.81929
[40]	Jegerlehner, F.; Kalmykov, M. Yu.; Veretin, O., Full two-loop electroweak corrections to the pole masses of gauge bosons, Nucl. Phys. B (Proc. Suppl.), 116, 382-386, (2003) · Zbl 1097.81929
[41]	Jegerlehner, F.; Kalmykov, M. Yu.; Kniehl, B. A., On the difference between the pole and the \(\overline{\operatorname{MS}}\) masses of the top quark at the electroweak scale, Phys. Lett. B, 722, 123-129, (2013) · Zbl 1311.81224
[42]	Bezrukov, F.; Kalmykov, M. Yu.; Kniehl, B. A.; Shaposhnikov, M., Higgs boson mass and new physics, J. High Energy Phys., 1210, (2012)
[43]	Sperling, M.; Stöckinger, D.; Voigt, A., Renormalization of vacuum expectation values in spontaneously broken gauge theories, J. High Energy Phys., 1307, (2013) · Zbl 1342.81201
[44]	Sperling, M.; Stöckinger, D.; Voigt, A., Renormalization of vacuum expectation values in spontaneously broken gauge theories: two-loop results, J. High Energy Phys., 1401, (2014)
[45]	Awramik, M.; Czakon, M.; Onishchenko, A.; Veretin, O., Bosonic corrections to δr at the two-loop level, Phys. Rev. D, 68, (2003)
[46]	Sirlin, A., Radiative corrections in the SU(2)_{L}×U(1) theory: A simple renormalization framework, Phys. Rev. D, 22, 971-981, (1980)
[47]	Buras, A. J., Weak Hamiltonian, CP violation and rare decays, (Probing the Standard Model of Particle Interactions. Proceedings, Summer School in Theoretical Physics, NATO Advanced Study Institute, 68th session, Les Houches, France, July 28-September 5, 1997. Pt. 1, 2, (1998)), 281-539, URL
[48]	Jegerlehner, F.; Kalmykov, M., \(\mathcal{O}(\alpha \alpha_s)\) relation between pole- and \(\overline{\operatorname{MS}}\)-mass of the t-quark, Acta Phys. Pol. B, 34, 5335-5344, (2003)
[49]	Kniehl, B. A.; Pikelner, A. F.; Veretin, O. L., Two-loop electroweak threshold corrections in the standard model, Nucl. Phys. B, 896, 19-51, (2015) · Zbl 1331.81372
[50]	Gray, N.; Broadhurst, D. J.; Grafe, W.; Schilcher, K., Three-loop relation of quark \(\overline{\operatorname{MS}}\) and pole masses, Z. Phys. C, 48, 673-679, (1990)
[51]	Avdeev, L. V.; Kalmykov, M. Yu., Pole masses of quarks in dimensional reduction, Nucl. Phys. B, 502, 419-435, (1997)
[52]	Fleischer, J.; Jegerlehner, F.; Tarasov, O. V.; Veretin, O. L.; Fleischer, J.; Jegerlehner, F.; Tarasov, O. V.; Veretin, O. L., Two-loop QCD corrections of the massive fermion propagator, Nucl. Phys. B, Nucl. Phys. B, 571, 511-512, (2000), (Erratum) · Zbl 0956.81054
[53]	Erler, J., Calculation of the QED coupling \(\hat{\alpha}(M_Z)\) in the modified minimal-subtraction scheme, Phys. Rev. D, 59, (1999)
[54]	Chetyrkin, K. G.; Kniehl, B. A.; Steinhauser, M., Decoupling relations to \(\mathcal{O}(\alpha_s^3)\) and their connection to low-energy theorems, Nucl. Phys. B, 510, 61-87, (1998)
[55]	Liu, T.; Steinhauser, M., Decoupling of heavy quarks at four loops and effective Higgs-fermion coupling, Phys. Lett. B, 746, 330-334, (2015)
[56]	Chetyrkin, K.; Kühn, J. H.; Steinhauser, M., : a Mathematica package for running and decoupling of the strong coupling and quark masses, Comput. Phys. Commun., 133, 43-65, (2000) · Zbl 0970.81087
[57]	Chetyrkin, K. G.; Kühn, J. H.; Maier, A.; Maierhöfer, P.; Marquard, P.; Steinhauser, M.; Sturm, C., Charm and bottom quark masses: an update, Phys. Rev. D, 80, (2009)
[58]	Baikov, P. A.; Chetyrkin, K. G.; Kühn, J. H., Quark mass and field anomalous dimensions to \(\mathcal{O}(\alpha_s^5)\), J. High Energy Phys., 1410, (2014)
[59]	Baikov, P. A.; Chetyrkin, K. G.; Kühn, J. H., Five-loop running of the QCD coupling constant · Zbl 1378.81150
[60]	Kniehl, B. A.; Pikelner, A. F.; Veretin, O. L., : a C++ library for the matching and running of the standard model parameters, Comput. Phys. Commun., 206, 84-96, (2016)
[61]	Mihaila, L. N.; Salomon, J.; Steinhauser, M., Gauge coupling beta functions in the standard model to three loops, Phys. Rev. Lett., 108, (2012)
[62]	Bednyakov, A. V.; Pikelner, A. F.; Velizhanin, V. N., Yukawa coupling beta-functions in the standard model at three loops, Phys. Lett. B, 722, 336-340, (2013) · Zbl 1306.81390
[63]	Chetyrkin, K. G.; Zoller, M. F., β-function for the Higgs self-interaction in the standard model at three-loop level, J. High Energy Phys., 1304, (2013) · Zbl 1380.81438
[64]	Bednyakov, A. V.; Pikelner, A. F., Four-loop strong coupling beta-function in the standard model, Phys. Lett. B, 762, 151-156, (2016)
[65]	Zoller, M. F., Top-Yukawa effects on the β-function of the strong coupling in the SM at four-loop level, J. High Energy Phys., 1602, (2016)
[66]	Bednyakov, A. V.; Kniehl, B. A.; Pikelner, A. F.; Veretin, O. L., Stability of the electroweak vacuum: gauge independence and advanced precision, Phys. Rev. Lett., 115, (2015) · Zbl 1356.81211
[67]	Bardin, D. Yu.; Vilenskiĭ, B. M.; Khristov, P. Kh., Calculation of the decay width of the Higgs boson. fermion decay modes, Sov. J. Nucl. Phys., Yad. Fiz., 53, 240-250, (1991)
[68]	Kniehl, B. A., Radiative corrections for \(H \rightarrow f \overline{f}(\gamma)\) in the standard model, Nucl. Phys. B, 376, 3-28, (1992)
[69]	Dabelstein, A.; Hollik, W., Electroweak corrections to the fermionic decay width of the standard Higgs boson, Z. Phys. C, 53, 507-516, (1992)
[70]	Kataev, A. L., Corrections of order \(\mathcal{O}(\overline{\alpha} \overline{\alpha}_s)\) and \(\mathcal{O}(\overline{\alpha}^2)\) to the \(\overline{b} b\)-decay width of the neutral Higgs boson, JETP Lett., 66, 327-330, (1997)
[71]	Mihaila, L.; Schmidt, B.; Steinhauser, M., \(\operatorname{\Gamma}(H \rightarrow b \overline{b})\) to order \(\alpha \alpha_s\), Phys. Lett. B, 751, 442-447, (2015)
[72]	Gorishny, S. G.; Kataev, A. L.; Larin, S. A.; Surguladze, L. R., Three-loop QCD correction to the correlator of the quark scalar currents and \(\operatorname{\Gamma}_{\operatorname{tot}}(H^0 \rightarrow \text{hadrons})\), Mod. Phys. Lett. A, 5, 2703-2711, (1990)
[73]	Chetyrkin, K. G.; Kühn, J. H.; Steinhauser, M., Heavy quark current correlators to \(\mathcal{O}(\alpha_s^2)\), Nucl. Phys. B, 505, 40-64, (1997)
[74]	Chetyrkin, K. G., Correlator of the quark scalar currents and \(\operatorname{\Gamma}_{\operatorname{tot}}(H \rightarrow \text{hadrons})\) at \(\mathcal{O}(\alpha_s^3)\) in pqcd, Phys. Lett. B, 390, 309-317, (1997)
[75]	Chetyrkin, K. G.; Steinhauser, M., Complete QCD corrections of order \(\mathcal{O}(\alpha_s^3)\) to the hadronic Higgs decay, Phys. Lett. B, 408, 320-324, (1997)
[76]	Baikov, P. A.; Chetyrkin, K. G.; Kühn, J. H., Scalar correlator at \(\mathcal{O}(\alpha_s^4)\), Higgs boson decay into bottom quarks, and bounds on the light-quark masses, Phys. Rev. Lett., 96, (2006)
[77]	Kniehl, B. A.; Spira, M., Two-loop \(\mathcal{O}(\alpha_s G_F m_t^2)\) correction to the \(H \rightarrow b \overline{b}\) decay rate, Nucl. Phys. B, 432, 39-48, (1994)
[78]	Chetyrkin, K. G.; Kniehl, B. A.; Steinhauser, M., Virtual top-quark effects on the \(H \rightarrow b \overline{b}\) decay at next-to-leading order in QCD, Phys. Rev. Lett., 78, 594-597, (1997)
[79]	Chetyrkin, K. G.; Kniehl, B. A.; Steinhauser, M., Three-loop \(\mathcal{O}(\alpha_s^2 G_F M_t^2)\) corrections to hadronic Higgs decays, Nucl. Phys. B, 490, 19-39, (1997)
[80]	Surguladze, L. R., \(\mathcal{O}(\alpha^n \alpha_s^m)\) corrections in \(e^+ e^-\) annihilation and τ decay
[81]	Mihaila, L., Three-loop gauge beta function in non-simple gauge groups, PoS, RADCOR 2013, (2013)
[82]	Chetyrkin, K. G., Quark mass anomalous dimension to \(\mathcal{O}(\alpha_s^4)\), Phys. Lett. B, 404, 161-165, (1997)
[83]	Vermaseren, J. A.M.; Larin, S. A.; van Ritbergen, T., The 4-loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B, 405, 327-333, (1997)
[84]	Chetyrkin, K. G.; Steinhauser, M., The relation between the \(\overline{\operatorname{MS}}\) and the on-shell quark mass at order \(\alpha_s^3\), Nucl. Phys. B, 573, 617-651, (2000)
[85]	Melnikov, K.; van Ritbergen, T., The three-loop relation between the \(\overline{\operatorname{MS}}\) and the pole quark masses, Phys. Lett. B, 482, 99-108, (2000)